Accelerating the Solution of Sequences of Large-Scale Linear Systems Using Randomized Subspace Projection

Core Concepts

A novel randomized subspace projection method is proposed to efficiently generate initial guesses for iteratively solving sequences of large-scale linear systems, leading to significant computational savings.

Abstract

The content discusses an algorithm for accelerating the solution of sequences of large-scale linear systems that arise, for example, from the discretization of time-dependent partial differential equations (PDEs).
The key highlights are:

The algorithm leverages the history of previously computed solutions to generate an initial guess for the current linear system, using a combination of reduced-order projection and randomized linear algebra techniques.

The randomized approach drastically reduces the computational cost compared to existing POD-based methods, while maintaining similar accuracy in the initial guess.

A convergence analysis is provided, showing that the accuracy of the initial guess improves rapidly as the size of the solution history increases.

The algorithm is applied to the simulation of turbulent plasma dynamics in a fusion device, leading to a significant reduction in the time required to solve the linear systems.

Stats

The number of GMRES iterations per timestep is reduced by a factor of 2.9 on average when using the proposed randomized subspace acceleration method, compared to the baseline approach using the previous solution as initial guess.
The computational time per timestep is reduced by a factor of 6.5 on average when using the randomized acceleration method, taking into account the cost of computing the initial guess.

Quotes

"The randomized method gives an acceleration comparable to the existing POD one, but it requires a much lower computational cost."
"The time employed by the POD method has not been included since it is significantly higher than the baseline, as predicted by the analysis in Section 2.1."

Key Insights Distilled From

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

by Margherita G... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2309.02156.pdf

Deeper Inquiries

How can the proposed randomized subspace acceleration method be extended to handle time-dependent coefficients in the linear systems that do not satisfy the analyticity assumption made in the convergence analysis

The proposed randomized subspace acceleration method can be extended to handle time-dependent coefficients in the linear systems that do not satisfy the analyticity assumption by incorporating adaptive strategies for handling non-analytic coefficients. One approach could involve utilizing data-driven techniques, such as machine learning algorithms, to adaptively adjust the subspace generation process based on the behavior of the time-dependent coefficients. By incorporating adaptive learning mechanisms, the method can dynamically adjust to the non-analytic behavior of the coefficients, ensuring efficient convergence even in the presence of non-analyticity.

Can the randomized approach be combined with other subspace recycling techniques, such as GCROT and GMRES-DR, to further improve the efficiency of the solver

Yes, the randomized approach can be combined with other subspace recycling techniques, such as GCROT and GMRES-DR, to further enhance the efficiency of the solver. By integrating randomized techniques with subspace recycling methods, the solver can benefit from the advantages of both approaches. For example, the randomized approach can be used to generate initial guesses for the subspace recycling methods, improving the quality of the recycled subspaces and accelerating the convergence of the iterative solver. This hybrid approach can leverage the strengths of each method to achieve even greater efficiency in solving sequences of large-scale linear systems.

What are the potential applications of the proposed method beyond the simulation of plasma dynamics, e.g., in other fields involving the solution of sequences of large-scale linear systems

The proposed randomized subspace acceleration method has potential applications beyond the simulation of plasma dynamics in various fields involving the solution of sequences of large-scale linear systems. Some potential applications include:

Computational Fluid Dynamics (CFD): The method can be applied to CFD simulations for solving time-dependent Navier-Stokes equations, where efficient solution of linear systems is crucial for accurate and timely simulations.
Structural Mechanics: In structural analysis and design, the method can be used to accelerate the solution of linear systems arising from finite element simulations, enabling faster and more accurate structural analysis.
Optimization: In optimization problems that involve solving sequences of linear systems, such as in iterative optimization algorithms, the method can improve the convergence speed and efficiency of the optimization process.
Climate Modeling: The method can be applied to climate modeling simulations that involve solving large-scale linear systems with time-dependent coefficients, aiding in the efficient and accurate prediction of climate patterns and trends.